Abstract
Based on zeroing neural network (ZNN), this paper designs two nonlinear activated ZNN (NAZNN) models for time-varying linear matrix equation through taking two new activation functions into consideration. The purpose of constructing the novel models is to solve the problem of time-varying linear matrix equation quickly and precisely. Theoretical analysis proves that two new activation functions can not only accelerate the convergence rate of the prime ZNN models but also come true finite-time convergence. After adding differential error and model-implementation error into the models, the theoretical upper bounds of the steady state residual errors are calculated, which demonstrate the superior robustness of the proposed two NAZNN models. Finally, comparative simulation results show the excellent performance of the proposed two NAZNN models by solving time-varying linear matrix equation.
Highlights
Solving linear matrix equation is a basic problem in many scientific and engineering fields
After the activation functions New Finite-Time Activation Function 1 (NFTAF1) and New Finite-Time Activation Function 2 (NFTAF2) are applied to the models, we prove the finite-time convergence of NAZNN1 and NAZNN2 models
In order to show the robustness of the NAZNN1 and NAZNN2 models, the differential error ∆d (t) ∈ R2×2 and the model-implement error ∆m (t) ∈ R2×2 are added into these two models, and they are set as the following form:
Summary
Solving linear matrix equation (matrix inversion can be regarded as a special linear matrix equation) is a basic problem in many scientific and engineering fields. Many gradient-related methods [16]–[19] are studied to solve the algebraic equation before ZNN These methods cannot work well when meeting with dynamic problems with time-varying coefficients. Li et al [27] proposed a nonlinear activation function which can realize finite-time convergence of ZNN models. We design two nonlinear activated ZNN (NAZNN) models (called NAZNN1 and NAZNN2) for specific time-varying linear matrix equation problem. 1) Two nonlinear activation functions are applied to the ZNN model to achieve finite-time convergence, and the theoretical upper bounds of the convergent time are calculated. 2) Two nonlinear activated ZNN models (i.e., NAZNN1 and NAZNN2) are developed to solve the time-varying linear matrix equations. 4) The comparative simulation results illustrate the correctness of relative theoretical analysis about convergence and robustness of the NAZNN1 and NAZNN2 models
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